Bayesian reasoning with ifs and ands and ors

Abstract

The Bayesian approach to the psychology of reasoning generalizes binary logic, extending the binary concept of consistency to that of coherence, and allowing the study of deductive reasoning from uncertain premises. Studies in judgment and decision making have found that people's probability judgments can fail to be coherent. We investigated people's coherence further for judgments about conjunctions, disjunctions and conditionals, and asked whether their coherence would increase when they were given the explicit task of drawing inferences. Participants gave confidence judgments about a list of separate statements (the statements group) or the statements grouped as explicit inferences (the inferences group). Their responses were generally coherent at above chance levels for all the inferences investigated, regardless of the presence of an explicit inference task. An exception was that they were incoherent in the context known to cause the conjunction fallacy, and remained so even when they were given an explicit inference. The participants were coherent under the assumption that they interpreted the natural language conditional as it is represented in Bayesian accounts of conditional reasoning, but they were incoherent under the assumption that they interpreted the natural language conditional as the material conditional of elementary binary logic. Our results provide further support for the descriptive adequacy of Bayesian reasoning principles in the study of deduction under uncertainty.

Keywords: coherence; conditionals; conjunction fallacy; deduction; uncertain reasoning.

FIGURE 1

Observed versus chance coherence for the six inferences of Experiment 1, (A) for the statements and (B) for the inferences task. Inferences 1 and 2 are logically equivalent or-introduction inferences, with a negation absent in 1 and present in the premise of 2. Inferences 3 and 4 are logically equivalent if-to-or inferences. Inference 3 has a negation in the conclusion and inference 4 in the premise. Inferences 5 and 6 are logically equivalent or-to-if inferences. Inference 5 has a negation in the conclusion and inference 6 in the premise. See Table 1 for the precise logical form of the inferences. Error bars show 95% CI.

FIGURE 2

Observed versus chance coherence for the four inferences of Experiment 2, (A) for the statements and (B) for the inferences task. Inferences 1 and 2 are and-to-if inferences. The first has the conjunction p and q as single premise, the second has p and q as two separate premises. Inferences 3 and 4 are and-elimination inferences. The first has prototypical, and the second counter-prototypical content for the scenario. See Table 1 for the precise logical form of the inferences. Error bars show 95% CI.